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Theorem 16: Every is semisimple. If is loxodromic, then is isometric to and acts on as translation by .

In particular, for every , is a -invariant subtree of .

Exercise 21: If and , then is loxodromic, , and .

(Hint: It is enough to construct an axis for .)

Lemma 20 (Helly’s Theorem for Trees): If are closed subtrees and for every , then .

Proof. Case : Let where . Let be the center of the triangle with vertices . Then .

General case: Let for . Then by the case. Now, by induction,

Corollary: If a finitely generated group acts on a tree with no global fixed point, then there is a (unique) -invariant subtree that is minimal with respect to inclusion, among all -invariant subtrees. Furthermore, is countable.

Proof. Let

For any -invariant , it is clear that . Therefore is minimal. Let be a finite generating set for . Suppose that every element of is elliptical, so for all , is elliptical. Then for all ,

Therefore by Lemma 20, a contradiction.

Suppose that are loxodromic and . By Exercise 21, intersects and non-trivially. Thus, is connected.

It remains to show that is -invariant. Let . For any ,

This implies that , and

So we conclude that is -invariant.

Definition: If is connected, then the graph of groups carried by is the graph of groups with underlying graph such that the vertex is labelled by , and the edge is labelled (with obvious edge maps). There is a natural map , where and . This map is an injection by the Normal Form Theorem.

Lemma 21: If is countable and is finitely generated, then there is a finite subgraph such that (where is the graph of groups carried by ).

Proof. Let be an exhaustion of by finite connected subgraphs. Let denote the graph of groups carried by and set . Since is finitely generated, there is an such that contains each generator of , and since , we conclude that .

Lemma 22: If is countable and is finitely generated then there is a finite minimal (wrt inclusion) subgraph that carries .

Proof. Let be the Bass-Serre tree of . Let . It’s an easy exercise to check that this is as required.

We will refer to (and , the graph of groups carried by ) as the core of .

Theorem 17: If is finitely generated, then decomposes as of a finite graph of groups . The vertex groups of are conjugate into the vertex groups of . The edge groups are likewise.

Fact: There exists a finitely generated non-Hopf group. (An example is the Baumslag-Solitar group , although we cannot prove it yet.) So, by Lemma 5, there is a finitely generated non-residually finite group. Thus, free groups are not ERF: if is a finitely generated non-residually finite group, then Lemma 4 implies that the kernel of a surjection is not separable in . However, finitely generated subgroups of free groups are separable:

Marshall Hall’s Theorem (1949): is LERF.

This proof is associated with Stallings.

Proof: As usual, let where is a rose. Let be a covering map with finitely generated. Let be compact. We need to embed in an intermediate finite-sheeted covering.

Enlarging if necessary, we may assume that is connected and that . Note that we have . By Theorem 5 (see below), the immersion extends to a covering . Then . So lifts to a map .

The main tool in the proof above is this:

Theorem 5: The immersion can be completed to a finite-sheeted covering into which embeds:

Proof: Color and orient the edges of . Any combinatorial map of graphs corresponds uniquely to a coloring and orientation on the edges of . A combinatorial map is an immersion if and only if at every vertex of , we see each color arriving at most once and leaving at most once. Likewise, it’s a covering map if and only if at each vertex, we see each color arriving exactly once and leaving exactly once.

Let be the number of vertices of . For each color , let be the number of edges of colored . Then there are vertices of missing “arriving” edges colored , and there are vertices of missing “leaving” edges colored . Choose any bijection between these two sets and use this to glue in edges colored . When this is done for all colors, the resulting map is clearly a covering.

Note that the proof in fact gives us more. For instance:

Exercise 6: If is a finitely generated subgroup of , then is a free factor of a finite-index subgroup of .

Exercise 7 (Greenberg’s Theorem): If and is finitely generated, then is of finite index in .